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Page 1
Edurev123
2. Shortest Distance between Two Skew
lines
2.1 Find the shortest distance between the lines
?? -?? ?? =
?? -?? ?? =?? -?? and ?? -???? =?? =
?? . For what value of ?? will the two lines intersect?
(2016 : 10 Marks)
Solution:
Lines are :
?? 1
·
?? -1
2
=
?? -2
4
=
?? -3
1
?? 2
:
?? 1
=
?? ?? =
?? 0
[
?? -???? =0
?? =0
]
?? 1
(1,2,3) on ?? 1
;?? 2
(0,0,0) on ?? 2
Shortest distance (SD) is the projection of ?? 1
?? 2
on ???? which is perpendicular to both
lines. Direction ratio's of ???? :
|
?? ?? ?? 2 4 1
1 ?? 0
| =??(0-?? )-?? (0-1)+?? (2?? -4)
=-???? +?? +(2?? -4)?? ???? =
1
v?? 2
+1+(2?? -4)
2
[-?? (1-0)+1(2-0)+(2?? -4)(3-0)
=|
5?? -10
v5?? 2
-16?? +17
|
The lines will intersect if, ???? =0, i.e. 5?? -10=0??? =2.
Page 2
Edurev123
2. Shortest Distance between Two Skew
lines
2.1 Find the shortest distance between the lines
?? -?? ?? =
?? -?? ?? =?? -?? and ?? -???? =?? =
?? . For what value of ?? will the two lines intersect?
(2016 : 10 Marks)
Solution:
Lines are :
?? 1
·
?? -1
2
=
?? -2
4
=
?? -3
1
?? 2
:
?? 1
=
?? ?? =
?? 0
[
?? -???? =0
?? =0
]
?? 1
(1,2,3) on ?? 1
;?? 2
(0,0,0) on ?? 2
Shortest distance (SD) is the projection of ?? 1
?? 2
on ???? which is perpendicular to both
lines. Direction ratio's of ???? :
|
?? ?? ?? 2 4 1
1 ?? 0
| =??(0-?? )-?? (0-1)+?? (2?? -4)
=-???? +?? +(2?? -4)?? ???? =
1
v?? 2
+1+(2?? -4)
2
[-?? (1-0)+1(2-0)+(2?? -4)(3-0)
=|
5?? -10
v5?? 2
-16?? +17
|
The lines will intersect if, ???? =0, i.e. 5?? -10=0??? =2.
2.2 Find the shortest distance between the skew lines,
?? -?? ?? =
?? -?? ?? =
?? -?? ?? and
?? +?? -?? =
?? +?? ?? =
?? -?? ?? .
(2017 : 10 Marks)
Solution:
Shortest distance lies along a direction which is perpendicular to both lines and given by
the cross-product of vectors along given two lines, ?? 1
,?? 2
.
??? =|
?? ?? ?? 3 -1 1
-3 2 4
|
=??(-4-2)-?? (12+3)+?? (6-3)
=-6?? -15?? +3?? =-3(2?? +5?? -?? )
? ??ˆ =
1
v30
(2?? +5?? -?? )
S.D. is the projection of ???? along ??ˆ.
???? =????
?????
·??ˆ
=
1
v30
[13-(-3))2+(8-(-7))5-(3-6)·1]
=
1
v30
(12+75+3)=
90
v30
=3v30
2.3 Find the shortest distance between the lines
Page 3
Edurev123
2. Shortest Distance between Two Skew
lines
2.1 Find the shortest distance between the lines
?? -?? ?? =
?? -?? ?? =?? -?? and ?? -???? =?? =
?? . For what value of ?? will the two lines intersect?
(2016 : 10 Marks)
Solution:
Lines are :
?? 1
·
?? -1
2
=
?? -2
4
=
?? -3
1
?? 2
:
?? 1
=
?? ?? =
?? 0
[
?? -???? =0
?? =0
]
?? 1
(1,2,3) on ?? 1
;?? 2
(0,0,0) on ?? 2
Shortest distance (SD) is the projection of ?? 1
?? 2
on ???? which is perpendicular to both
lines. Direction ratio's of ???? :
|
?? ?? ?? 2 4 1
1 ?? 0
| =??(0-?? )-?? (0-1)+?? (2?? -4)
=-???? +?? +(2?? -4)?? ???? =
1
v?? 2
+1+(2?? -4)
2
[-?? (1-0)+1(2-0)+(2?? -4)(3-0)
=|
5?? -10
v5?? 2
-16?? +17
|
The lines will intersect if, ???? =0, i.e. 5?? -10=0??? =2.
2.2 Find the shortest distance between the skew lines,
?? -?? ?? =
?? -?? ?? =
?? -?? ?? and
?? +?? -?? =
?? +?? ?? =
?? -?? ?? .
(2017 : 10 Marks)
Solution:
Shortest distance lies along a direction which is perpendicular to both lines and given by
the cross-product of vectors along given two lines, ?? 1
,?? 2
.
??? =|
?? ?? ?? 3 -1 1
-3 2 4
|
=??(-4-2)-?? (12+3)+?? (6-3)
=-6?? -15?? +3?? =-3(2?? +5?? -?? )
? ??ˆ =
1
v30
(2?? +5?? -?? )
S.D. is the projection of ???? along ??ˆ.
???? =????
?????
·??ˆ
=
1
v30
[13-(-3))2+(8-(-7))5-(3-6)·1]
=
1
v30
(12+75+3)=
90
v30
=3v30
2.3 Find the shortest distance between the lines
?? ?? ?? +?? ?? ?? +?? ?? ?? +?? ?? =?? ?? ?? ?? +?? ?? ?? +?? ?? ?? +?? ?? =??
and ?????? ?? -axis.
(2018 : 12 Marks)
Solution:
The equation of ?? -axis is ?? =?? =0
? Any plane, ?? , through ?? -axis can be written as
??¨+???? =0 (??)
Further, any plane ?? 2
, through given set of planes is
?? 1
?? +?? 1
?? +?? 1
?? +?? 1
+?? (?? 2
?? +?? 2
?? +?? 2
?? +?? 2
)=0
i.e., (?? 1
+?? ?? 2
)?? +(?? 1
+?? ?? 2
)?? +(?? 1
+?? ?? 2
)?? +?? 1
+?? ?? 2
=0 (???? )
For shortest distance ?? 1
and ?? 2
should be parallel.
?
?? 1
+?? ?? 2
1
=
?? 1
+???
?? 2
?? =
?? 1
+?? ?? 2
0
i.e., ?? 1
+?? ?? 2
=0
? ?? =
?? 1
?? 2
? equation of ?? 2
is
(?? 1
-
?? 1
?? 2
?? 2
)?? +(?? 1
-
?? 1
?? 2
?? 2
)?? +(?? 1
-
?? 1
?? 2
)?? 2
=0
Shortest distance,
?? =
|?? 1
+?? ?? 2
-0|
v(?? 1
+?? ?? 2
)
2
+(?? 1
+?? ?? 2
)
2
+0
2
?? =
|?? 2
?? 1
+?? 1
?? 2
|
v(?? 2
?? 1
-?? 1
?? 2
)
2
+(?? 2
?? 1
-?? 1
?? 2
)
2
2.4 Show that the lines
?? +?? -?? =
?? -?? ?? =
?? +?? ?? and
?? ?? =
?? -?? -?? =
?? +?? ?? intersect. Find the
coordinates of the point of intersection and the equation of the plane containing
them.
(2019: 10 Marks)
Page 4
Edurev123
2. Shortest Distance between Two Skew
lines
2.1 Find the shortest distance between the lines
?? -?? ?? =
?? -?? ?? =?? -?? and ?? -???? =?? =
?? . For what value of ?? will the two lines intersect?
(2016 : 10 Marks)
Solution:
Lines are :
?? 1
·
?? -1
2
=
?? -2
4
=
?? -3
1
?? 2
:
?? 1
=
?? ?? =
?? 0
[
?? -???? =0
?? =0
]
?? 1
(1,2,3) on ?? 1
;?? 2
(0,0,0) on ?? 2
Shortest distance (SD) is the projection of ?? 1
?? 2
on ???? which is perpendicular to both
lines. Direction ratio's of ???? :
|
?? ?? ?? 2 4 1
1 ?? 0
| =??(0-?? )-?? (0-1)+?? (2?? -4)
=-???? +?? +(2?? -4)?? ???? =
1
v?? 2
+1+(2?? -4)
2
[-?? (1-0)+1(2-0)+(2?? -4)(3-0)
=|
5?? -10
v5?? 2
-16?? +17
|
The lines will intersect if, ???? =0, i.e. 5?? -10=0??? =2.
2.2 Find the shortest distance between the skew lines,
?? -?? ?? =
?? -?? ?? =
?? -?? ?? and
?? +?? -?? =
?? +?? ?? =
?? -?? ?? .
(2017 : 10 Marks)
Solution:
Shortest distance lies along a direction which is perpendicular to both lines and given by
the cross-product of vectors along given two lines, ?? 1
,?? 2
.
??? =|
?? ?? ?? 3 -1 1
-3 2 4
|
=??(-4-2)-?? (12+3)+?? (6-3)
=-6?? -15?? +3?? =-3(2?? +5?? -?? )
? ??ˆ =
1
v30
(2?? +5?? -?? )
S.D. is the projection of ???? along ??ˆ.
???? =????
?????
·??ˆ
=
1
v30
[13-(-3))2+(8-(-7))5-(3-6)·1]
=
1
v30
(12+75+3)=
90
v30
=3v30
2.3 Find the shortest distance between the lines
?? ?? ?? +?? ?? ?? +?? ?? ?? +?? ?? =?? ?? ?? ?? +?? ?? ?? +?? ?? ?? +?? ?? =??
and ?????? ?? -axis.
(2018 : 12 Marks)
Solution:
The equation of ?? -axis is ?? =?? =0
? Any plane, ?? , through ?? -axis can be written as
??¨+???? =0 (??)
Further, any plane ?? 2
, through given set of planes is
?? 1
?? +?? 1
?? +?? 1
?? +?? 1
+?? (?? 2
?? +?? 2
?? +?? 2
?? +?? 2
)=0
i.e., (?? 1
+?? ?? 2
)?? +(?? 1
+?? ?? 2
)?? +(?? 1
+?? ?? 2
)?? +?? 1
+?? ?? 2
=0 (???? )
For shortest distance ?? 1
and ?? 2
should be parallel.
?
?? 1
+?? ?? 2
1
=
?? 1
+???
?? 2
?? =
?? 1
+?? ?? 2
0
i.e., ?? 1
+?? ?? 2
=0
? ?? =
?? 1
?? 2
? equation of ?? 2
is
(?? 1
-
?? 1
?? 2
?? 2
)?? +(?? 1
-
?? 1
?? 2
?? 2
)?? +(?? 1
-
?? 1
?? 2
)?? 2
=0
Shortest distance,
?? =
|?? 1
+?? ?? 2
-0|
v(?? 1
+?? ?? 2
)
2
+(?? 1
+?? ?? 2
)
2
+0
2
?? =
|?? 2
?? 1
+?? 1
?? 2
|
v(?? 2
?? 1
-?? 1
?? 2
)
2
+(?? 2
?? 1
-?? 1
?? 2
)
2
2.4 Show that the lines
?? +?? -?? =
?? -?? ?? =
?? +?? ?? and
?? ?? =
?? -?? -?? =
?? +?? ?? intersect. Find the
coordinates of the point of intersection and the equation of the plane containing
them.
(2019: 10 Marks)
Solution:
Any point on the line
?? +1
-3
=
?? -3
2
=
?? +2
1
is (-1-3?? ,3+2?? ,-2+
?? ) …(??)
Similarly, any part on the line
?? 1
=
?? -7
-3
=
?? +7
2
is (?? 2
,7-3?? 1
,-7+
2?? -1
) …(???? )
If the two given lines intersect then for some value of ?? and ?? 2
the two above points (i)
and (ii) must coincide. i.e.,
-1-3?? =?? 1
3+2?? =7-3?? 1
-2+?? =-7+2?? -1
Solving the first two of these equations, we get
?? =-1,?? 1
=2
The so values of ?? and ?? ' satisfy the third equation also. Hence, the given lines intersect.
Substituting these values ?? and ?? in (1) or (2) we get the required coordinates of the
point of intersection as (2,1,-3) ,
Also, the equation of the plane containing the given lines is
|
?? +1 ?? -3 ?? +2
-3 2 1
1 -3 2
|=0
?(?? +1)(4+3)-(?? -3)(-6-1)+(?? +2)(9-2)
=0
?? +?? +?? =0
which is the required equation.
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Jan 05, 2025
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